It's not just cedar balls that can damage fabric. Anything made from wood fibers, including cardboard boxes, contains acid that gradually eats into fabric. "Archival quality" papers and cardboards have been treated to remove the acid, so they are safe, but they are also horribly expensive.
In short, you do not want to store fabric in contact with paper, cardboard, or wood of any kind (including cedar chests). It's okay to store fabric or quilts in a cedar chest *if* they are encased so the fabric does not have direct contact with the wood.